Answer
$$\frac{{81}}{{10}}$$
Work Step by Step
$$\eqalign{
& \int_{ - 1}^2 {x\left( {1 + {x^3}} \right)} dx \cr
& {\text{mutliply}} \cr
& \int_{ - 1}^2 {\left( {x + {x^4}} \right)} dx \cr
& {\text{find the antiderivative by the power rule}} \cr
& = \left( {\frac{{{x^2}}}{2} + \frac{{{x^{4 + 1}}}}{{4 + 1}}} \right)_{ - 1}^2 \cr
& = \left( {\frac{{{x^2}}}{2} + \frac{{{x^5}}}{5}} \right)_{ - 1}^2 \cr
& {\text{part 1 of fundamental theorem of calculus}} \cr
& = \left( {\frac{{{{\left( 2 \right)}^2}}}{2} + \frac{{{{\left( 2 \right)}^5}}}{5}} \right) - \left( {\frac{{{{\left( { - 1} \right)}^2}}}{2} + \frac{{{{\left( { - 1} \right)}^5}}}{5}} \right) \cr
& {\text{simplify}} \cr
& = \left( {\frac{{42}}{5}} \right) - \left( {\frac{3}{{10}}} \right) \cr
& = \frac{{81}}{{10}} \cr} $$
